1/38 Tagged Data Tag: a symbol in a data structure that identifies its type Why we need tags Extended example: evaluating arithmetic expressions

Generic Operations 2/38 NAME SALARY JOB RECORD TYPE 1 RECORD TYPE 2 RECORD TYPE 3 … GENERIC OPERATIONS

3/38 Manipulating complex numbers Complex number has: real and imaginary part (Cartesian) magnitude and angle (polar) (define (+c z1 z2) (make-rectangular (+ (real-part z1)(real-part z2)) (+ (imag-part z1)(imag-part z2)))) (define (*c z1 z2) (make-polar (* (magnitude z1) (magnitude z2)) (+ (angle z1) (angle z2)))) magnitude angle Real part Imag part Addition is easier in Cartesian coordinates Multiplication is easier in polar coordinates

4/38 Bert’s data structure (define (make-rectangular rl im) (cons rl im)) (define (make-polar mg an) (cons (* mg (cos an)) (* mg (sin an)))) (define (real-part cx) (car cx)) (define (imag-part cx) (cdr cx)) (define (magnitude cx) (sqrt (+ (square (real cx)) (square (imag cx))))) (define (angle cx) (atan (imag cx) (real cx))) Note conversion to rectangular form before storing Need to do some computation since stored in rectangular form

5/38 Ernie’s data structure (define (make-rectangular rl im) (cons (sqrt (+ (square rl) (square im))) (atan im rl))) (define (make--polar mg an) (cons mg an)) (define (real-part cx) (* (mag cx) (cos (angle cx)))) (define (imag-part cx) (* (mag cx) (sin (angle cx)))) (define (magnitude cx) (car cx)) (define (angle cx) (cdr cx)) Note conversion to polar form before storing Need to do some computation since stored in polar form

6/38 Whose number is it? Suppose we pick up the following object 12 What number does this represent? Bert Ernie

Typed Data 7/38 TYPECONTENTS (define (attach-type type contents) (cons type contents)) (define (type datum) (car datum)) (define (contents datum) (cdr datum)) (define (rectangular? z) (eq? (type z) ‘rectangular)) (define (polar? z) (eq? (type z) ‘polar))

Rectangular Package (define (make-rectangular x y) (attach-type ‘rectangular (cons x y))) (define (real-part-rectangular z) (car z)) (define (imag-part-rectangular z) (cdr z)) (define (magnitude-rectangular z) (sqrt (+ (square (car z)) (square (cdr z))))) (define (angle-rectangular z) (atan (cdr z) (car z))) 8/38

Polar Package (define (make-polar r a) (attach-type ‘polar (cons r a)) (define (real-part-polar z) (* (car z) (cos (cdr z)))) (define (imag-part-polar z) (* (car z) (sin (cdr z)))) (define (magnitude-polar z) (car z)) (define (angle-polar z) (cdr z)) 9/38

Generic Selection from Complex Numbers (define (REAL-PART z) (cond ((rectangular? z) … ) ((polar? z) … ))) (define (IMAG-PART z) (cond ((rectangular? z) … ) ((polar? z) … ))) (define (MAGNITUDE z) (cond ((rectangular? z) … ) ((polar? z) … ))) (define (ANGLE z) (cond ((rectangular? z) … ) ((polar? z) … ))) 10/38 (define (real-part z) (cond ((rectangular? z) (real-part-rectangular (contents z))) ((polar? z) (real-part-polar (contents z))))) …

Complex Number 11/38 RECTPOLAR REALIMAGMAGANGLE +C-C*C/C

Operation Table 12/38 RECTPOLAR REAL-PART-POLAR REAL-PART-RECT MAGNITUDE-POLARMAGNITUDE-RECT IMAG-PART-POLARIMAG-PART-RECT ANGLE-POLARANGLE-RECT REAL-PART IMAG-PART MAGNITUDE ANGLE (PUT KEY1 KEY2 VALUE) (GET KEY1 KEY2)

Install Operations in the table Installing the rectangular operations in the table (put ‘rectangular ‘real-part real-part-rectangular) (put ‘rectangular ‘imag-part imag-part-rectangular) (put ‘rectangular ‘magnitude magnitude-rectangular) (put ‘rectangular ‘angle angle-rectangular) Installing the polar operations in the table (put ‘polar ‘real-part real-part-polar) (put ‘polar ‘imag-part imag-part-polar) (put ‘polar ‘magnitude magnitude-polar) (put ‘polar ‘angle angle-polar) 13/38

Dispatch on Type – Data Directed Programming (define (operate op obj) (let ((proc (get (type obj) op))) (if (not (null? proc)) (proc (contents obj)) (error “undefined operation”)))) (define (real-part obj) (operate ‘real-part obj)) (define (imag-part obj) (operate ‘imag-part obj)) (define (magnitude obj) (operate ‘magnitude obj)) (define (angle obj) (operate ‘angle obj)) 14/38

Generic Arithmetic System 15/38 RECTPOLAR COMPLEX +C-C *C/C ADD SUB MUL DIV RATIOANL +RAT *RAT ORDINARY NUMS +- */

Rational Number (define (+ rat x y) (make-rat (+ (* (numer x) (denom y)) (* (denom x) (numer y))) (* (denom x) (denom y)))) (define (-rat x y) …) (define (*rat x y) …) (define (/rat x y) …) (define (make-rat x y) (attach-type ‘rational (cons x y))) (put ‘rational ‘add +rat) (put ‘rational ‘sub -rat) (put ‘rational ‘mul *rat) (put ‘rational ‘div /rat) 16/38

operate-2 (define (ADD x y) (OPERATE-2 ‘ADD x y)) (define (operate-2 op arg1 arg2) (if (eq? (type arg1) (type arg2)) (let ((proc (get (type arg1) op))) (if (not (null? proc)) (proc (contents arg1) (contents arg2)) (error “op: undefined on type”))) (error “args not same type”))) 17/38

Installing Complex Number 18/38 RECTPOLAR COMPLEX +C-C *C/C ADD SUB MUL DIV RATIOANL +RAT *RAT ORDINARY NUMS +- */ +COMPLEX -COMPLEX (define (make-complex z) (attach-type ‘complex z)) (define (+complex z1 z2) (make-complex (+c z1 z2))) (put ‘complex ‘add +complex) similarly for -complex, *complex, /complex

Complex Number Example 19/38 (3 + 4i) * (2+6i) (define (MUL x y) (OPERATE-2 ‘MUL x y)) RECT34 COMPLEX TYPE for +C -C *C /CTYPE for ADD SUB MUL DIV MUL

Installing Polynomials X X  ((15 1) (7 2) (0 5)) (POLYNOMIAL X ) (define (make-polynomial var term-list) (attach-type ‘polynomial (cons var term-list))) (define (+poly p1 p2) (if (same-var? (var p1) (var p2)) (make-polynomial (var p1) (+terms (term-list p1) (term-list p2))) (error “Polys not in same var”))) (put ‘polynomial ‘add +poly) 20/38

Installing Polynomials (define (+terms L1 L2) (cond ((empty-termlist? L1) L2) ((empty-termlist? L2) L1) (else (let ((t1 (first-term L1)) (t2 (first-term L2))) (cond ((> (order t1) (order t2)) … ) ((< (order t1) (order t2)) … ) (else … )))))) 21/38

Installing Polynomials ((> (order t1) (order t2) (adjoin-term t1 (+terms (rest-terms L1) L2))) ((< (order t1) (order t2) (adjoin-term t2 (+term L1 (rest-terms L2)))) (else (adjoin-terms (make-term (order t1) (ADD (coeff t1) (coeff t2))) (+terms (rest-terms L1) (rest-terms L2)))) 22/38

Generic Arithmetic System 23/38 RECTPOLAR COMPLEX +C -C *C/C ADD SUB MUL DIV RATIOANL +RAT *RAT ORDINARY NUMBERS +- */ +COMPLEX -COMPLEX POLYNOMIALS +POLY … RECTPOLAR COMPLEX +C -C *C/C RATIOANL +RAT *RAT ORDINARY NUMERS +- */ +COMPLEX -COMPLEX POLY.. … (define (+ rat x y) (make-rat (ADD (MUL (numer x) (denom y)) (MUL (denom x) (numer y))) (MUL (denom x) (denom y))))

24/38 Benefits of tagged data data-directed programming: functions that decide what to do based on argument types example: in a graphics program area: triangle|square|circle -> number defensive programming: functions that fail gracefully if given bad arguments –much better to give an error message than to return garbage!

25/38 Example: Arithmetic evaluation (define an-expr (make-sum (make-sum 3 15) 20)) an-expr ==> (+ (+ 3 15) 20) (eval an-expr) ==> 38 Expressions might include values other than simple numbers Ranges: some unknown number between min and max arithmetic: [3,7] + [1,3] = [4,10] Limited precision values: some value  some error amount arithmetic: (100  1) + (3  0.5) = (103  1.5) Create arithmetic expressions Evaluate arithmetic expressions to reduce them to simpler form

26/38 Approach: start simple, then extend Characteristic of all software engineering projects Start with eval for numbers, then add support for ranges and limited-precision values Goal: build eval in a way that it will extend easily & safely Easily: requires data-directed programming Safely: requires defensive programming Process: multiple versions of eval eval-1 Simple arithmetic, no tags eval-2 Extend the evaluator, observe bugs eval-3 through -7 Do it again with tagged data

27/38 1. Data abstraction for sums of numbers (define (make-sum addend augend) ; type: Exp, Exp -> SumExp (list '+ addend augend)) (define (sum-exp? e) ; type: anytype -> boolean (and (pair? e) (eq? (first e) '+))) (define (sum-addend sum) (second sum)) (define (sum-augend sum) (third sum)) ; type: SumExp -> Exp the type Exp will be different in different versions of eval

28/38 1. Eval for sums of numbers ; Exp = number | SumExp (define (eval-1 exp) ; type: Exp -> number (cond ((number? exp) exp) ((sum-exp? exp) (+ (eval-1 (sum-addend exp)) (eval-1 (sum-augend exp)))) (else (error "unknown expression " exp)))) (eval-1 (make-sum 4 (make-sum 3 5))) ==> 12 ; base case ; recursive case

29/38 Example in gory detail (eval-1 (make-sum 4 (make-sum 3 5))) ==> Sum-exp? checks this using eq? (+ (eval-1 ) (eval-1 )).. Number? checks this (+ 4 ) Sum-exp? checks this using eq? (+ (eval-1 ) (eval-1 )).. (+ 4 (+ 3 5))

30/38 2. Extend the abstraction to ranges (without tags) ; type: number, number -> range2 (define (make-range-2 min max) (list min max)) ; type: range2 -> number (define (range-min-2 range) (first range)) (define (range-max-2 range) (second range)) ; type: range2, range2 -> range2 (define (range-add-2 r1 r2) (make-range-2 (+ (range-min-2 r1) (range-min-2 r2)) (+ (range-max-2 r1) (range-max-2 r2))))

31/38 Detailed example of adding ranges 3713 (make-range-2 (+ ) (+ )) (make-range ) 4 10 This is a range (range-add-2 (make-range 3 7) (make-range 1 3))

32/38 2. Eval for sums of numbers and ranges (broken!) ; Exp = number | range2 | SumExp (define (eval-2 exp) ; type: Exp -> number|range2 (cond ((number? exp) exp) ((sum-exp? exp) (let ((v1 (eval-2 (sum-addend exp))) (v2 (eval-2 (sum-augend exp)))) (if (and (number? v1) (number? v2)) (+ v1 v2) ; add numbers (range-add-2 v1 v2)))) ; add ranges ((pair? exp) exp) ; a range (else (error "unknown expression " exp))))

33/38 Why is eval-2 broken? Missing a case: sum of number and a range (eval-2 (make-sum 4 (make-range-2 4 6))) ==> error: the object 4 is not a pair

34/38 2. Eval for sums of numbers and ranges (broken!) ; Exp = number|range2|SumExp (define (eval-2 exp) ; type: Exp -> number|range2 (cond ((number? exp) exp) ((sum-exp? exp) (let ((v1 (eval-2 (sum-addend exp))) (v2 (eval-2 (sum-augend exp)))) (if (and (number? v1) (number? v2)) (+ v1 v2) ; add numbers (range-add-2 v1 v2)))) ; add ranges ((pair? exp) exp) ; a range (else (error "unknown expression " exp)))) Range-add-2 expects two ranges, i.e. two lists!

35/38 Why is eval-2 broken? Missing a case: sum of number and a range (eval-2 (make-sum 4 (make-range-2 4 6))) ==> error: the object 4 is not a pair Not defensive: what if we add limited-precision values but forget to change eval-2 ? (define (make-limited-precision-2 val err) (list val err)) (eval-2 (make-sum (make-range-2 4 6) (make-limited-precision ))) ==> (14 7) correct answer: (13 17) or (15 2) Key point – doesn’t return an error, but gives us what appears to be a legitimate answer!

36/38 Lessons from eval-2 Common bug: calling a function on the wrong type of data typos brainos changing one part of the program and not another Common result: the function returns garbage Why? Primitive predicates like number? and pair? are ambiguous Something fails later, but cause is hard to track down Worst case: program produces incorrect output!! Next: how to use tagged data to ensure that the program halts immediately

37/38 3. Start again using tagged data Take another look at SumExp... it's already tagged! (define sum-tag '+) ; Type: Exp, Exp -> SumExp (define (make-sum addend augend) (list sum-tag addend augend)) ; Type: anytype -> boolean (define (sum-exp? e) (and (pair? e) (eq? (first e) sum-tag))) sum-exp? is not ambiguous: only true for things made by make-sum (assuming the tag + isn't used anywhere else)

38/38 Data abstraction for numbers using tags (define constant-tag 'const) ; type: number -> ConstantExp (define (make-constant val) (list constant-tag val)) ; type: anytype -> boolean (define (constant-exp? e) (and (pair? e) (eq? (first e) constant-tag))) ; type: ConstantExp -> number (define (constant-val const) (second const))

39/38 3. Eval for numbers with tags (incomplete) ; Exp = ConstantExp | SumExp (define (eval-3 exp) ; type: Exp -> number (cond ((constant-exp? exp) (constant-val exp)) ((sum-exp? exp) (+ (eval-3 (sum-addend exp)) (eval-3 (sum-augend exp)))) (else (error "unknown expr type: " exp) ))) Not all nontrivial values used in this code are tagged (eval-3 (make-sum (make-constant 3) (make-constant 5))) ==> 8 No closure!

40/38 4. Eval for numbers with tags ; type: Exp -> ConstantExp (define (eval-4 exp) (cond ((constant-exp? exp) exp) ((sum-exp? exp) (make-constant (+ (constant-val (eval-4 (sum-addend exp))) (constant-val (eval-4 (sum-augend exp))) ))) (else (error "unknown expr type: " exp)))) (eval-4 (make-sum (make-constant 3) (make-constant 5))) ==> (constant 8) There is that pattern of using selectors to get parts, doing something, then using constructor to reassemble

41/38 Make add an operation in the Constant abstraction ;type: ConstantExp,ConstantExp -> ConstantExp (define (constant-add c1 c2) (make-constant (+ (constant-val c1) (constant-val c2)))) ; type: ConstantExp | SumExp -> ConstantExp (define (eval-4 exp) (cond ((constant-exp? exp) exp) ((sum-exp? exp) (constant-add (eval-4 (sum-addend exp)) (eval-4 (sum-augend exp)))) (else (error "unknown expr type: " exp))))

42/38 Lessons from eval-3 and eval-4 standard pattern for a data abstration with tagged data a variable stores the tag attach the tag in the constructor write a predicate that checks the tag –determines whether an object belongs to the type of the abstraction operations strip the tags, operate, attach the tag again must use tagged data everywhere to get full benefits including return values

43/38 5. Same pattern: ranges with tags (define range-tag 'range) ; type: number, number -> RangeExp (define (make-range min max) (list range-tag min max)) ; type: anytype -> boolean (define (range-exp? e) (and (pair? e) (eq? (first e) range-tag))) ; type: RangeExp -> number (define (range-min range) (second range)) (define (range-max range) (third range)) [3, 7]

44/38 5. Eval for numbers and ranges with tags ; Exp = ConstantExp | RangeExp | SumExp (define (eval-5 exp) ; type: Exp -> ConstantExp|RangeExp (cond ((constant-exp? exp) exp) ((range-exp? exp) exp) ((sum-exp? exp) (let ((v1 (eval-5 (sum-addend exp))) (v2 (eval-5 (sum-augend exp)))) (if (and (constant-exp? v1) (constant-exp? v2)) (constant-add v1 v2) (range-add (val2range v1) (val2range v2))))) (else (error "unknown expr type: " exp))))

45/38 Simplify eval with a data-directed add function ; ValueExp = ConstantExp | RangeExp (define (value-exp? v) (or (constant-exp? v) (range-exp? v))) ; type: ValueExp, ValueExp -> ValueExp (define (value-add-6 v1 v2) (if (and (constant-exp? v1) (constant-exp? v2)) (constant-add v1 v2) (range-add (val2range v1) (val2range v2)))) ; val2range: if argument is a range, return it ; else make the range [x x] from a constant x ; This is called coercion

46/38 Use type coercion to turn constants into ranges (define (val2range val) ; type: ValueExp -> RangeExp (if (range-exp? val) val ; just return range (make-range (constant-val val) (constant-val val))))

47/38 6. Simplified eval for numbers and ranges ; ValueExp = ConstantExp | RangeExp ; Exp = ValueExp | SumExp (define (eval-6 exp) ; type: Exp -> ValueExp (cond ((value-exp? exp) exp) ((sum-exp? exp) (value-add-6 (eval-6 (sum-addend exp)) (eval-6 (sum-augend exp)))) (else (error "unknown expr type: " exp))))

48/38 Compare eval-6 with eval-1 (define (eval-6 exp) (cond ((value-exp? exp) exp) ((sum-exp? exp) (value-add-6 (eval-6 (sum-addend exp)) (eval-6 (sum-augend exp)))) (else (error "unknown expr type: " exp)))) Compare to eval-1. It is just as simple! (define (eval-1 exp) (cond ((number? exp) exp) ((sum-exp? exp) (+ (eval-1 (sum-addend exp)) (eval-1 (sum-augend exp)))) (else (error "unknown expression " exp)))) This shows the power of data-directed programming

49/38 Eval-7: adding limited-precision numbers (define limited-tag 'limited) (define (make-limited-precision val err) (list limited-tag val err)) ; Exp = ValueExp | Limited | SumExp (define (eval-7 exp) ; type: Exp -> ValueExp | Limited (cond ((value-exp? exp) exp) ((limited-exp? exp) exp) ((sum-exp? exp) (value-add-6 (eval-7 (sum-addend exp)) (eval-7 (sum-augend exp)))) (else (error "unknown expr type: " exp)))) 5 +/- 2

50/38 Oops: value-add-6 is not defensive (eval-7 (make-sum (make-range 4 6) (make-limited-precision 10 1))) ==> (range 14 16) WRONG (define (value-add-6 v1 v2) (if (and (constant-exp? v1) (constant-exp? v2)) (constant-add v1 v2) (range-add (val2range v1) (val2range v2)))) Correct answer should have been (range 13 17) or (limited 15 2)

51/38 What went wrong in value-add-6 ? limited-exp is not a constant, so falls into the alternative (limited 10 1) passed to val2range (limited 10 1) passed to constant-val, returns 10 range-add called on (range 4 6) and (range 10 10) (define (value-add-6 v1 v2) (if (and (constant-exp? v1) (constant-exp? v2)) (constant-add v1 v2) (range-add (val2range v1) (val2range v2)))) (define (val2range val) (if (range-exp? val) val ; just return range (make-range (constant-val val) ; assumes constant (constant-val val))))

52/38 7. Defensive version: check tags before operating ; type: ValueExp, ValueExp -> ValueExp (define (value-add-7 v1 v2) (cond ((and (constant-exp? v1) (constant-exp? v2)) (constant-add v1 v2)) ((and (value-exp? v1) (value-exp? v2)) (range-add (val2range v1) (val2range v2))) (else (error "unknown exp: " v1 " or " v2)))) Rule of thumb: when checking types, use the else branch only for errors

53/38 Lessons from eval-5 through eval-7 Data directed programming can simplify higher level code Using tagged data is only defensive programming if you check the tags don't put code in the else branch of if or cond ; make it signal an error instead Traditionally, operations and accessors don't check tags They assume tags have been checked at the higher level A check in constant-val would have trapped this bug Be paranoid: add checks in your operations and accessors –The cost of redundant checks is usually trivial compared to the cost of your debugging time Andy Grove: "only the paranoid survive"